MAT244--2019F > Chapter 9

differences between proper and improper nodes

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Richard Qiu:
Hello guys, could anyone help me to explain the differences between proper and improper nodes? btw, any suggestions on how to remember the types and stability of the critical points?

Amanda-fazi:
Since both proper and improper nodes have equal eigenvalues, the differences between these two nodes is that: proper node/star point has two independent eigenvectors, while improper/degenerate node has only one independent eigenvector by (A-rI)x =0, and we create a generalized eigenvector associated with the repeated eigenvalues by letting (A-rI)y = x.

Amanda-fazi:
There are mainly 5 cases of Eigenvalues(from book Elementary Differential Equations and Boundary Value Problems-11th Edition section 9.1):
as it is mentioned above, the equal eigenvalues case mentioned above is CASE 3.

CASE 1: Real, Unequal Eigenvalues of the Same Sign
CASE 2: Real Eigenvalues of Opposite Sign                            ->saddle point
CASE 3: Equal Eigenvalues
CASE 4: Complex Eigenvalues with Nonzero Real Part
CASE 5: Pure Imaginary Eigenvalues                                     ->center

After memorized there are five cases, CASE 1, CASE 3 and CASE 4 have two branches while the rest of the cases(CASE 2 and CASE 5) only have one:
to be more specific:

CASE 1: Real, Unequal Eigenvalues of the Same Sign separated into:
a)lambda1 >lambda2 >0:
critical point called node/nodal source
a)lambda1 <lambda2 <0:
critical point called node/nodal sink

CASE 3:Equal Eigenvalues separated into:
a)two independent eigenvectors:
critical point called proper node or star point
b)one independent eigenvector:
critical point called improper node or degenerate node

CASE 4:Complex Eigenvalues with Nonzero Real Part separated into:
a)pointing-outward trajectories as lambda > 0:
critical point called spiral source
a)pointing-inward trajectories as lambda < 0:
critical point called spiral sink

For the stability, as long as there is one lambda>0, then it is unstable, and the last one lambda=0 is stable. For the rest of them, asymptotically stable applied.

anntara khan:
I made this handy color coded guide to help me remember all the cases:

anntara khan:
Based on the stability near locally linear system I have extended the previously posted table, hope this helps remembering :)